On the Complexity of Computing Generators of Closed Sets

  • Miki Hermann
  • Barış Sertkaya
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4933)


We investigate the computational complexity of some decision and counting problems related to generators of closed sets fundamental in Formal Concept Analysis. We recall results from the literature about the problem of checking the existence of a generator with a specified cardinality, and about the problem of determining the number of minimal generators. Moreover, we show that the problem of counting minimum cardinality generators is Open image in new window -complete. We also present an incremental-polynomial time algorithm from relational database theory that can be used for computing all minimal generators of an implication-closed set.


Relational Database Minimal Generator Truth Assignment Formal Context Formal Concept Analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Armstrong, W.W.: Dependency structures of data base relationships. In: Rosenfeld, J.L. (ed.) Proceedings 6th Information Processing Conference (IFIP 1974), Stockholm, Sweden, pp. 580–583. North-Holland, Amsterdam (1974)Google Scholar
  2. 2.
    Durand, A., Hermann, M., Kolaitis, P.G.: Subtractive reductions and complete problems for counting complexity classes. Theoretical Computer Science 340(3), 496–513 (2005)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Frambourg, C., Valtchev, P., Godin, R.: Merge-based computation of minimal generators. In: Dau, F., Mugnier, M.-L., Stumme, G. (eds.) ICCS 2005. LNCS (LNAI), vol. 3596, pp. 181–194. Springer, Heidelberg (2005)Google Scholar
  4. 4.
    Ganter, B.: Two basic algorithms in concept analysis. Technical Report Preprint-Nr. 831, Technische Hochschule Darmstadt, Germany (1984)Google Scholar
  5. 5.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (1999)MATHGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  7. 7.
    Guigues, J.-L., Duquenne, V.: Familles minimales d’implications informatives resultant d’un tableau de données binaries. Mathématiques, Informatique et Sciences Humaines 95, 5–18 (1986)MathSciNetGoogle Scholar
  8. 8.
    Gunopulos, D., et al.: Discovering all most specific sentences. ACM Transactions on Database Systems 28(2), 140–174 (2003)CrossRefGoogle Scholar
  9. 9.
    Hemaspaandra, L.A., Vollmer, H.: The satanic notations: Counting classes beyond #P and other definitional adventures. SIGACT News, Complexity Theory Column 8 26(1), 2–13 (1995)CrossRefGoogle Scholar
  10. 10.
    Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Information Processing Letters 27(3), 119–123 (1988)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kuznetsov, S.O.: On computing the size of a lattice and related decision problems. Order 18(4), 313–321 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kuznetsov, S.O.: On the intractability of computing the Duquenne-Guigues base. Journal of Universal Computer Science 10(8), 927–933 (2004)MathSciNetGoogle Scholar
  13. 13.
    Kuznetsov, S.O., Obiedkov, S.A.: Comparing performance of algorithms for generating concept lattices. Journal of Experimental and Theoretical Artificial Intelligence 14(2-3), 189–216 (2002)MATHCrossRefGoogle Scholar
  14. 14.
    Kuznetsov, S.O., Obiedkov, S.O.: Counting pseudo-intents and #P-completeness. In: Missaoui, R., Schmidt, J. (eds.) Formal Concept Analysis. LNCS (LNAI), vol. 3874, pp. 306–308. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Lucchesi, C.L., Osborn, S.L.: Candidate keys for relations. Journal of Computer and System Science 17(2), 270–279 (1978)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Maier, D.: The Theory of Relational Databases. Computer Science Press (1983)Google Scholar
  17. 17.
    Nehmé, K., et al.: On computing the minimal generator family for concept lattices and icebergs. In: Ganter, B., Godin, R. (eds.) ICFCA 2005. LNCS (LNAI), vol. 3403, pp. 192–207. Springer, Heidelberg (2005)Google Scholar
  18. 18.
    Osborn, S.L.: Normal Forms for Relational Data Bases. PhD thesis, University of Waterloo, Canada (1977)Google Scholar
  19. 19.
    Papadimitriou, C.H.: Computational complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  20. 20.
    Toda, S., Watanabe, O.: Polynomial-time 1-Turing reductions from #PH to #P. Theoretical Computer Science 100(1), 205–221 (1992)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8(3), 410–421 (1979)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Valtchev, P., Missaoui, R., Godin, R.: Formal concept analysis for knowledge discovery and data mining: The new challenges. In: Eklund, P.W. (ed.) ICFCA 2004. LNCS (LNAI), vol. 2961, pp. 352–371. Springer, Heidelberg (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Miki Hermann
    • 1
  • Barış Sertkaya
    • 2
  1. 1.LIX (CNRS, UMR 7161), École Polytechnique, 91128 PalaiseauFrance
  2. 2.Institut für Theoretische Informatik, TU DresdenGermany

Personalised recommendations